Descending Chain Condition Research Articles

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

A module satisfying the descending chain condition on cyclic submodules is called coperfect. The class of coperfect modules lies properly between the class of locally artinian modules and the class of semiartinian modules. Let R be a commutative ring with identity. We show that every semiartinian R-module is coperfect if and only if R is a T-ring. It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if 𝔪/𝔪2 is a finitely generated R-module for every maximal ideal 𝔪 of R.

EQUATIONAL THEORIES OF FIELDS

AbstractA first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields and of the theory of separably closed fields of arbitrary imperfection degree.

Some remarks on the volume of log varieties

AbstractIn this note, using methods introduced by Haconet al. [‘Boundedness of varieties of log general type’, Proceedings of Symposia in Pure Mathematics, Volume 97 (American Mathematical Society, Providence, RI, 2018) 309–348], we study the accumulation points of volumes of varieties of log general type. First, we show that if the set of boundary coefficientsΛsatisfies the descending chain condition (DCC), is closed under limits and contains 1, then the corresponding set of volumes satisfies the DCC and is closed under limits. Then, we consider the case ofε-log canonical varieties, for 0 < ε < 1. In this situation, we prove that ifΛis finite, then the corresponding set of volumes is discrete.

Projectivity and linkage for completely join irreducible ideals of an expanded group

Specialized from lattice theory to the lattice of ideals of an expanded group, the equivalence relation of projectivity between two intervals of ideals I[A, B] and I[C, D] of an expanded group V yields an isomorphism between B / A and D / C. Conversely, E. Aichinger has shown that if V is finite and B / A and D / C are minimal factors of V such that B / A and D / C are isomorphic as $$C_0(V)$$-modules where $$C_0(V)$$ is the nearring of functions on V that are 0-preserving and congruence preserving, then I[A, B] and I[C, D] are projective. Restricting this equivalence relation to intervals of the form $$I[U^-,U]$$ where U is a completely join irreducible ideal of V and $$U^-$$ is the unique ideal of V maximal in U plays a central role in this work. In this paper we introduce another equivalence relation involving the completely join irreducible ideals of V called linkage. We explore this linkage equivalence relation and its relationship to projectivity for completely join irreducible ideals. As a byproduct of our studies we will extend Aichinger’s result from the finite case to the one where $$C_0(V)$$ satisfies a finiteness condition called the weak descending chain condition on right ideals previously studied by the authors.

On Bernstein algebras satisfying chain conditions II

Following a previous work with Boudi, we continue to investigate Bernstein algebras satisfying chain conditions. First, it is shown that a Bernstein algebra A with ascending or descending chain condition on subalgebras is finite-dimensional. We also prove that A is Noetherian (Artinian) if and only if its barideal is. Next, as a generalization of Jordan and nuclear Bernstein algebras, we study whether a Noetherian (Artinian) Bernstein algebra A with a locally nilpotent barideal N is finite-dimensional. The response is affirmative in the Noetherian case, unlike in the Artinian case. This question is closely related to a result by Zhevlakov on general locally nilpotent nonassociative algebras that are Noetherian, for which we give a new proof. In particular, we derive that a commutative nilalgebra of nilindex 3 which is Noetherian or Artinian is finite-dimensional. Finally, we improve and extend some results of Micali and Ouattara to the Noetherian and Artinian cases.

On the ascending and descending chain conditions in the lattice of monoid varieties

In this work we consider monoids as algebras with an associative binary operation and the nullary operation that fixes the identity element. We found an example of two varieties of monoids with finite subvariety lattices such that their join covers one of them and has a continuum cardinality subvariety lattice that violates the ascending chain condition and the descending chain condition.

On accumulation points of volumes of log surfaces

Let be a set satisfying the descending chain condition. We show that every accumulation point of volumes of log canonical surfaces with coefficients in can be realized as the volume of a log canonical surface with big and nef and with coefficients in in such a way that at least one coefficient lies in . As a corollary, if , then all accumulation points of volumes are rational numbers. This proves a conjecture of Blache. For the set of standard coefficients we prove that the minimal accumulation point is between and .

On the Boundedness of Anti-Canonical Volumes of Singular Fano3-Folds in Characteristicp> 5

AbstractIn this article, we prove the following version of the weak Borisov-Alexeev-Borisov (Weak-BAB) conjecture for $3$-folds in $\operatorname{char} p>5$: fix a set $I\subseteq [0, 1)$ which satisfies the descending chain condition (DCC) and an algebraically closed field $k$ of characteristic $p>5$. Let ${\mathfrak{D}}$ be a collection of klt pairs $(X, \Delta )$ satisfying the following properties: (1) $X$ is a projective $3$-fold, (2) $\Delta $ is an ${\mathbb{R}}$-divisor with coefficients in $I$, (3) $K_X+\Delta \equiv 0$, and (4) $-K_X$ is ample. Then the set $\{\operatorname{vol}_X(-K_X) \ | \ (X, \Delta )\in{\mathfrak{D}}\mbox{ for some }\Delta \}$ is bounded from above.

On groups where the twisted conjugacy class of the unit element is a subgroup

We study groups G where the -conjugacy class of the unit element is a subgroup of G for every automorphism of G. If G has n generators, then we prove that the k-th member of the lower central series has a finite verbal width bounded in terms of n, k. Moreover, we prove that if such group G satisfies the descending chain condition for normal subgroups, then G is nilpotent, what generalizes the result from [Bardakov, Nasybullov, and Neshchadim]. Finally, if G is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of G.

Singular and semi-simple character in E-injective N-groups with weakly descending chain conditions

Injective modules and near-ring groups have been studied by several researchers like Mason et al., Faith et al., Jara, Harada, Cheng. Of these Oswald and Mason have studied injective and projective near-ring modules. Mason studied injective near-ring modules and defined the concepts like n-injective, loosely injective and almost injective near-ring modules. In Hazarika and Saikia (Int J Math Sci 33(2), 2013) we extended the notion of relative injectivity of modules to near-ring groups. Here E-injective N-groups with descending chain conditions are studied. It is shown that the singular and semi-simple characters play a vital role in characterization of E-injective N-groups with weakly descending chain conditions.

Modules with Chain Conditions on S-Closed Submodules

Let L be a commutative ring with identity and let W be a unitary left L- module. A submodule D of an L- module W is called s- closed submodule denoted by D ≤sc W, if D has no proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D = H. In this paper, we study modules which satisfies the ascending chain conditions (ACC) and descending chain conditions (DCC) on this kind of submodules.

Modules with Chain Condition on Non-finitely Generated Submodules

In this article, we study modules with chain condition on non-finitely generated submodules. We show that if an R-module M satisfies the ascending chain condition on non-finitely generated submodules, then M has Noetherian dimension and its Noetherian dimension is less than or equal to one. In particular, we observe that if an R-module M satisfies the ascending chain condition on non-finitely generated submodules, then every submodule of M is countably generated. We investigate that if an R-module M satisfies the descending chain condition on non-finitely generated submodules, then M has Krull dimension and its Krull dimension may be any ordinal number $$\alpha $$ . In particular, if a perfect R-module M satisfies the descending chain condition on non-finitely generated submodules, then it is Artinian.

Uniquely Complemented Posets

We study complementation in bounded posets. It is known and easy to see that every complemented distributive poset is uniquely complemented. The converse statement is not valid, even for lattices. In the present paper we provide conditions that force a uniquely complemented poset to be distributive. For atomistic resp. atomic posets as well as for posets satisfying the descending chain condition we find sufficient conditions in the form of so-called LU-identities. It turns out that for finite posets these conditions are necessary and sufficient.

Spectral partially ordered sets

ABSTRACTWe examine some topics related to (gold)spectral partially ordered sets, i.e., those that are order isomorphic to (Goldman) prime spectra of commutative rings. Among other results, we characterize the partially ordered sets that are isomorphic to prime spectra of rings satisfying the descending chain condition on radical ideals, and we show that a dual of a tree is isomorphic to the Goldman prime spectrum of a ring if and only if every chain has an upper bound. We also give some new methods for constructing (gold)spectral partially ordered sets.

О СВОЙСТВАХ ПЕРВИЧНОГО РАДИКАЛА СЛАБОАРТИНОВОЙ АЛГЕБРЫ ЛИ

This article deals with the issues of the structural theory of Lie algebras. The construction of the structural theory of algebraic systems implies the existence of certain structures of a special form, which are simpler than the base system. The important tool to study algebraic systems is the radical. The development of the structural theory of Lie algebras led to the emergence of various radicals. There are many radicals of Lie algebras in numerous publications. For example, the Killing radical, the Parfenov radical, the Jacobson radical and the prime radical are considered in various articles. The important area of research is the study of radicals of innite-dimensional Lie algebras. The article is devoted to proving properties of prime radical of a weakly artinian Lie algebra. A Lie algebra is said to be a weakly artinian if the Lie algebra satises the descending chain condition on ideals. In the rst section of the paper we introduced the concept of the prime radical in the following way. A Lie algebra L is said to be prime if [U; V ] = 0 implies U = 0 or V = 0 for any ideals U and V of L. We say that the ideal P of a Lie algebra L is prime if the factor algebra L=P is prime. The intersection of all prime ideals is called the prime radical P(L) of a Lie algebra L. In the second section it is shown that any nite set of elements of the prime radical of a weakly artinian Lie algebra generates the nilpotent subalgebra. This means that the prime radical is locally nilpotent. The third section is devoted to the solvability of the prime radical of a weakly artinian Lie algebra. There is a history of solving Mikhalev' s problem about the prime radical of a weakly artinian Lie algebra in this section also.

Local orders in Jordan algebras

Making use of results on general algebras of quotients of Jordan algebras, we study a notion of local order based on the version for linear Jordan algebras of the ideas of Fountain and Gould [14] as adapted to the Jordan context by Fernández-López and García-Rus in [7]. In particular, we characterize the set of Lesieur–Croisot elements of a nondegenerate Jordan algebra as those elements of the Jordan algebra lying in the socle of its maximal algebra of quotients, and apply this relationship to extend to quadratic Jordan algebras the results of Fernández-López and García-Rus on local orders in nondegenerate Jordan algebras satisfying the descending chain condition on principal inner ideals and not containing ideals which are nonartinian quadratic factors.

Realizing posets as prime spectra of Leavitt path algebras

We associate in a natural way to any partially ordered set (P,≤) a directed graph EP (where the vertices of EP correspond to the elements of P, and the edges of EP correspond to related pairs of elements of P), and then describe the prime spectrum Spec(LK(EP)) of the resulting Leavitt path algebra LK(EP). This construction allows us to realize a wide class of partially ordered sets as the prime spectra of rings. More specifically, any partially ordered set in which every downward directed subset has a greatest lower bound, and where these greatest lower bounds satisfy certain compatibility conditions, can be so realized. In particular, any partially ordered set satisfying the descending chain condition is in this class.

On the structure of compact graphs

A simple graph \(G\) is called a compact graph if \(G\) contains no isolated vertices and for each pair \(x\), \(y\) of non-adjacent vertices of \(G\), there is a vertex \(z\) with \(N(x)\cup N(y)\subseteq N(z)\), where \(N(v)\) is the neighborhood of \(v\), for every vertex \(v\) of \(G\). In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph \(G\), then the descending chain condition holds for the set of neighbors of \(G\).

Rings with divisibility on chains of ideals

ABSTRACTWe present a generalization of the ascending and descending chain condition on one-sided ideals by means of divisibility on chains. We say that a ring R satisfies ACCd on right ideals if in every ascending chain of right ideals of R, each right ideal in the chain, except for a finite number of right ideals, is a left multiple of the following one; that is, each right ideal in the chain, except for a finite number, is divisible by the following one. We study these rings and prove some results about them. Dually, we say that a ring R satisfies DCCd on right ideals if in every descending chain of right ideals of R, each right ideal in the chain, except for a finite number of right ideals, is divisible by the previous one. We study these conditions on rings, in general and in special cases.

Local solvability of the prime radical of a weakly artinian Lie algebra

We prove the local solvability of the prime radical of a Lie algebra with descending chain condition for the ideals. We note that an analog of this statement holds also for the graded Ω-groups with finiteness condition.